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Chapter 4: Problem 7

If \(3 a+2 b=24\) and \(4 a+5 b=53\), what is the value of \(a+b\) ? A) 2 B) 7 C) 9 D) 11

Short Answer

Expert verified
(D) 11

Step by step solution

01

Write down the equations

We are given the following two equations: \( (1) \ 3a + 2b = 24 \) \( (2) \ 4a + 5b = 53 \)
02

Eliminate one of the variables

We want to eliminate one of the variables to get an equation with only one variable. Notice that if we multiply equation (1) by 2, we get the same coefficient for \(b\) as in equation (2). Multiply equation (1) by 2: \( 6a + 4b = 48 \) Now, subtract equation (2) from the new equation: \( (6a + 4b) - (4a + 5b) = 48 - 53 \) \( 2a - b = -5 \) We now have a new equation with variables \(a\) and \(b\): \( (3) \ 2a - b = -5 \)
03

Solve for one variable

Now, solve equation (3) for \(b\): \( b = 2a + 5 \)
04

Substitute the value of one variable in the other equation

Next, substitute the expression we found for \(b\) in equation (1): \( 3a + 2(2a + 5) = 24 \)
05

Solve for the remaining variable

Now, solve for \(a\): \( 3a + 4a + 10 = 24 \) \( 7a = 14 \) \( a = 2 \)
06

Substitute the value of the variable in the other equation

Now, substitute the value of \(a\) we found into the expression for \(b\): \( b = 2(2) + 5 \) \( b = 4 + 5 \) \( b = 9 \)
07

Calculate the value of a + b

Finally, calculate the value of \(a + b\): \( a + b = 2 + 9 = 11 \) The value of \(a + b\) is 11, so the correct answer is (D) 11.

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